If you have a feeling a limit exists you can show it does by using the definition for example. Usually simpler though is the squeeze theorem.

By the way, use brackets next time, the you've written it it looks like:
[tex]f(x,y)=\frac{xy}{|x|}+|y|[/tex]
and I guess it should be:
[tex]f(x,y)=\frac{xy}{|x|+|y|}[/tex]
or f(x,y)=xy/(|x|+|y|)
Also, you haven't stated the point of the limit. In this case, I guess it should be (0,0)

For iii:
You probably already found out that, if the limit exists, it must be zero. So try to find a function g(x,y) that goes to zero as (x,y)->(0,0) and for which [itex]|f(x,y)| \leq g(x,y)|[/itex]. Then, according to the squeeze theorem we have f(x,y)-> 0